The Lebesgue measure on $\mathbb{R}^d$ admits the following polar decomposition: $$ L(dx) = r^{d-1} dr \lambda(dy), $$ where 𝜆 is the uniform measure on the Euclidean unit sphere of $\mathbb{R}^d$ and where $x=ry$. Now, change the uniform measure on the Euclidean unit sphere by another positive finite measure (non-degenerate) on the Euclidean unit sphere. Namely, $$ L'(dx) = r^{d-1}dr\sigma(dy). $$ where $\sigma$ is a positive finite measure (non-degenerate) on the Euclidean unit sphere $\mathbb{R}^d$ and where $x=ry$. Where can I find harmonic analysis results based on this new reference measure ?